• Corpus ID: 125524017

Algorithms for Tverberg's theorem via centerpoint theorems

@article{Rolnick2016AlgorithmsFT,
  title={Algorithms for Tverberg's theorem via centerpoint theorems},
  author={David Rolnick and Pablo Sober{\'o}n},
  journal={arXiv: Computational Geometry},
  year={2016}
}
  • D. Rolnick, P. Soberón
  • Published 4 January 2016
  • Mathematics, Computer Science
  • arXiv: Computational Geometry
We obtain algorithms for computing Tverberg partitions based on centerpoint approximations. This applies to a wide range of convexity spaces, from the classic Euclidean setting to geodetic convexity in graphs. In the Euclidean setting, we present probabilistic algorithms which are weakly polynomial in the number of points and the dimension. For geodetic convexity in graphs, we obtain deterministic algorithms for cactus graphs and show that the general problem of finding the Radon number is NP… 
6 Citations

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References

SHOWING 1-10 OF 25 REFERENCES
Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets
TLDR
A new variation of Tverberg’s theorem is presented, given a discrete set S, and the number of points needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S.
Radon partitions in convexity spaces
TLDR
It is shown that the case k=2 does imply that every set of O(k^2 log^2 k) points admits a Tverberg partition into k parts, so that the convex hulls of the parts have a common intersection.
Optimal bounds for the colored Tverberg problem
We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al.
A Generalisation of Tverberg’s Theorem
We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝd of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A1,A2,…,Ak such that for any C⊂S of at most r
Quantitative Tverberg, Helly, & Carathéodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the
A Colored Version of Tverberg's Theorem
The main result of this paper is that given n red, n white, and n green points in the plane, it is possible to form n vertex-disjoint triangles Delta_{1},...,Delta_{n} in such a way that the
Approximate centerpoints with proofs
Inapproximability results for graph convexity parameters
PARTITION NUMBERS FOR TREES AND ORDERED SETS
In this paper some bounds on the Tveberg-type convexity partition numbers of abstract spaces will be presented. The main objective is to show that a conjecture of J. Eckhoff relating the Tverberg
Approximating center points with iterative Radon points
TLDR
The algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics and has the potential to improve results in practice for constructing weak ∊-nets.
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